# Part II: mathematical tools and physical principles

Part II includes Chapter 2 (Mathematical Tools) and Chapter 3 (Physical Principles). These chapters introduce the mathematical tools and physical principles required for Parts III and IV using motivational applications to particular geologic structures. For example, among many other applications:

- Chapter 2 introduces scalar, vector, and tensor quantities using the classic study by Ernst Cloos of deformed ooliths from the Paleozoic limestone of the South Mountain fold in Maryland and Pennsylvania; and
- Chapter 3 derives the equations for conservation of linear momentum using a fault that crops out in the Lake Edison granodiorite of the Sierra Nevada, California.

## Deformed ooliths approximate ellipsoids

Thin section images of ooliths from South Mountain fold. 2 millimeter scale bar. (a) Nearly undeformed ooliths with mean axial ratio = 1.16. (b) Deformed ooliths with mean axial ratio = 1.56. Modified from Cloos, 1971. The geometric shapes of the undeformed and deformed ooliths provide data that is necessary to quantify the deformation using a **tensor**. Because the deformed oolith approximates an *ellipsoid*, the stretch of material lines varies with orientation and the **deformation gradient tensor** quantifies the stretch in all possible orientations for a given oolith.

## Momentum conservation during faulting

Fault in Lake Edison Granodiorite, Sierra Nevada, CA. Pen 14 cm long. The small white square is fixed in space and granodiorite with a given density moves through this square with variable velocity, **v**, so momentum is carried in and out of the square as slip accelerates and then decelerates. The basic tenets of continuum mechanics do not admit the spontaneous creation or destruction of momentum, so momentum is conserved. This basic principle leads to Cauchy's equations of motion, which describe brittle, ductile, and viscous deformation at all length and time scales that are relevant to structural geology.